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Amplitwist

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Concept used to represent a derivative

In mathematics, the amplitwist is a concept created by Tristan Needham in the book Visual Complex Analysis (1997) to represent the derivative of a complex function visually.

Definition

The amplitwist associated with a given function is its derivative in the complex plane. More formally, it is a complex number z {\displaystyle z} such that in an infinitesimally small neighborhood of a point a {\displaystyle a} in the complex plane, f ( ξ ) = z ξ {\displaystyle f(\xi )=z\xi } for an infinitesimally small vector ξ {\displaystyle \xi } . The complex number z {\displaystyle z} is defined to be the derivative of f {\displaystyle f} at a {\displaystyle a} .

Uses

The concept of an amplitwist is used primarily in complex analysis to offer a way of visualizing the derivative of a complex-valued function as a local amplification and twist of vectors at a point in the complex plane.

Examples

Define the function f ( z ) = z 3 {\displaystyle f(z)=z^{3}} . Consider the derivative of the function at the point e i π 4 {\displaystyle e^{i{\frac {\pi }{4}}}} . Since the derivative of f ( z ) {\displaystyle f(z)} is 3 z 2 {\displaystyle 3z^{2}} , we can say that for an infinitesimal vector γ {\displaystyle \gamma } at e i π 4 {\displaystyle e^{i{\frac {\pi }{4}}}} , f ( γ ) = 3 ( e i π 4 ) 2 γ = 3 e i π 2 γ {\displaystyle f(\gamma )=3(e^{i{\frac {\pi }{4}}})^{2}\gamma =3e^{i{\frac {\pi }{2}}}\gamma } .

References

  1. ^ Tristan., Needham (1997). Visual complex analysis. Oxford: Clarendon Press. ISBN 0198534477. OCLC 36523806.
  2. Soto-Johnson, Hortensia; Hancock, Brent (February 2019). "Research to Practice: Developing the Amplitwist Concept". PRIMUS. 29 (5): 421–440. doi:10.1080/10511970.2018.1477889.
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